9. Let X be a continuous random variable taking values between 0 and 2 with probability...

    

Transcribed Image Text:

9. Let X be a continuous random variable taking values between 0 and 2 with probability density function p(x) = 0.5. Find E(X) and Var(X). Plot its Cumulative Distribution Function. 10. (5pt) Suppose that X and Y are two normally distributed random variables. X has mean 2 and standard deviation 1. Y has mean 1 and standard deviation 2. Their correlation is 0.5. What is the mean and standard deviation of X + Y ? What is the distribution of X+Y? What if X and Y are jointly normally distributed? What if they are not jointly normally distributed? Explain your answer. 11 (5pt) This is not a Math Finance question but a frequent interview question. What is the first time after 3pm when the hour and minute hands of a clock are exactly on top of each other. 12. (5pt) Suppose you are applying to graduate schools. Your chances to be admitted to each one school are 5% and are the same for any school. To how many different schools you need to apply if you want your chances to be admitted to at least one school to be above 95%. 9. Let X be a continuous random variable taking values between 0 and 2 with probability density function p(x) = 0.5. Find E(X) and Var(X). Plot its Cumulative Distribution Function. 10. (5pt) Suppose that X and Y are two normally distributed random variables. X has mean 2 and standard deviation 1. Y has mean 1 and standard deviation 2. Their correlation is 0.5. What is the mean and standard deviation of X + Y ? What is the distribution of X+Y? What if X and Y are jointly normally distributed? What if they are not jointly normally distributed? Explain your answer. 11 (5pt) This is not a Math Finance question but a frequent interview question. What is the first time after 3pm when the hour and minute hands of a clock are exactly on top of each other. 12. (5pt) Suppose you are applying to graduate schools. Your chances to be admitted to each one school are 5% and are the same for any school. To how many different schools you need to apply if you want your chances to be admitted to at least one school to be above 95%.